In this paper we study the effects of institutional constraints on stability, efficiency and network formation. More precisely, an exogenous 'societal cover' consisting of a collection of possibly overlapping subsets that covers the whole set of players and such that no set in this collection is contained in another specifies the social organization in different groups or 'societies'. It is assumed that a player may initiate links only with players that belong to at least one society that s/he also belongs to, thus restricting the feasible strategies and networks. In this way only the players in the possibly empty 'societal core', i.e., those that belong to all societies, may initiate links with all individuals. In this setting the part of the current network within each connected component of the cover is assumed to be common knowledge to all players in that component. Based on this two-ingredient model, network and societal cover, we examine the impact of societal constraints on stable/efficient architectures and on dynamics.